面向气象预报数值模式的高效并行计算研究
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摘要
大气科学尤其数值模式的发展进步与并行计算息息相关，这一点已经成为气象预报和并行计算两个领域的共识。随着并行计算技术的不断发展进步，并行计算机的规模愈来愈大，计算能力愈来愈强，这使得更为快捷、准确地进行数值预报成为可能。而预报模式能否高效地应用于高性能计算机，主要取决于模式所采用的数值方法是否可以实现有效的并行计算。
本文从气象预报数值模式分布式存储并行计算的实际需要出发，着眼于谱元素方法和有限差分的并行计算性能，研究了二维半隐谱元素浅水波模式、中国科学院大气物理研究所第三代海洋环流模式和中国新一代多尺度预报模式等三个气象预报模式的高效并行计算，设计实现了相应的应用软件。本文的主要研究成果概括如下：
(1) 讨论了数值并行计算技术的发展现状及其对数值预报模式的影响，深入研究了并行计算模型、并行编程模型和数值并行软件的软件工程等问题。
(2) 关于谱元素方法及两维半隐谱元素浅水波模式的并行计算
ⅰ) 依据切比契夫多项式和勒让德多项式的计算特点，提出了一个混合局部基函数预备定理，给出了两类混合局部基谱元素格式。采用混合局部基函数可以在不降低计算精度的情况下使元素矩阵具有特定的稀疏特征，且能保持谱元素方法在元素界面上的C~0连续性。
ⅱ) 将有限元方法中的逐元素(EBE)策略应用于谱元素方法的并行计算，在PC机群上设计实现了并行应用软件PSES。研究表明：即使在网络性能瓶颈非常突出的以太网PC机群上，采用高阶谱元素方法(高于7阶)和通常的空间离散规模(32×32个元素以上)求解两维椭圆问题，可实现60％以上的8机并行效率。
ⅲ) 研究了两维半隐谱元素浅水波模式的高效并行计算，给出了离散方程的逐元素(EBE)并行求解技巧，设计了相应的并行应用软件，并在PC机群上获得高效实现。
(3) 关于中科院大气物理研究所第三代海洋环流模式的高效并行计算
ⅰ) 基于对有限差分离散的分析，提出了一个面向显式时间积分并行计算的多重叠
国防科学技术大学研究生院学位论文
边界优化模型，指出了网络速度与处理器速度间的落差使得采用冗余计算取代
部分通信成为一种具有相对优势的实用技术。结合以矩阵矢量乘积为主的迭代
计算特点，从通信量、通信计算比和并行可扩展性等方面比较分析了离散格式
对于并行计算的影响。
11)发展了国内第一个高效并行海洋环流模式:深入研究了中国科学院大气物理所
第三代海洋环流模式，分析了其串行计算特点和并行可扩展性，指出了其计算
方法上存在的对并行计算的某些限制，对该模式组织了有效并行计算，并成功
应用了包括多重叠边界在内的若干并行优化方法，在国产小型MPP系统和SGI
originZoo。上都获得了较高的性能。
(4)关于中国新一代多尺度预报模式动力内核的高效计算问题
i)三维复杂核姆霍兹方程的离散求解是中国新一代多尺度预报模式动力内核中
的一个瓶颈问题，该问题的计算性能决定了整个数值模式的效率。对此，本文
提出了一种基于逐层门限技术的近似逆矩阵稀疏模式预选方法，并构造了相应
的稀疏近似逆预条件子，结合GCR算法和G州[R卫S算法，首次将逐层门限稀
疏近似逆预条件子应用于新一代多尺度预报模式动力内核的实际计算，数值实
验表明这里给出的方法可以大大提高数值模式的计算效率。
The development of atmosphere science, especially the numerical meteorologic forecasting model, and the parallel computing are closed linked. To meet the practical requirements for distributed-memory parallel computing of numerical forecasting models, we study on the parallel characteristics of spectral element and finite difference methods. On this basis, we develop fast and parallel computations for three meteorologic models, including 2-D shallow water equations, the new generation multi-scale weather forecasting model and the ocean general circulation model. The main work is summarized as follows:
(1) We analyze the situation of the parallel computing technology and it's influence on numerical forecasting models, and point out that the numerical models now are in serious demand for efficient parallel algorithms. We also review parallel computing model, parallel programming model and the software engineering for numerical applications.
(2) On spectral element method and parallel computing of 2-D spectral element shallow water model
i) We put forward a hybrid-local-basis theorem, and give two kinds of local basis schemes fitting for both the Chebyshev polynomials and the Legendre case. With these spectral element schemes, we can get elemental matrix with specific sparse pattern, without sacrificing precision. In addition, the famous C?continuity can be hold by these schemes.
ii) The parallel computing of spectral element methods by element-by-element strategy is analyzed deeply, and we design a high-performance spectral element parallel software package named PSES. Numerical tests show that the high-order (>7) spectral element methods can performance efficiently on the relatively slow-network PC clusters, and parallel efficiency with eight nodes is more than 60% with normal elemental scale (32*32 elements).
iii) For the semi-implicit spectral element method on the 2-D shallow water equations, we give a parallel mass-matrix-diagonal preconditioned CG method with the element-by-element strategy and develop the corresponding parallel software package. We also get high efficiency on PC clusters.
(3) On parallel computing of the third generation Ocean General Circulation Model from LASG/IAP
i) We present an optimization model by multi-overlapping-boundary for parallel explicit integration with finite difference discretization, and point out that the gap between speed of network and CPU makes it a practical technique. We compare the parallel characteristics of spectral element and finite difference from the point of view of communications, communication-to-computation ratio and scalability.
ii) We organize the parallel computing of'the third generation ocean general circulation model from the Chinese Institute of Atmosphere Physics (LASG/IAP), and we also apply several parallel optimization strategies including multi-overlapping-boundary. The parallelism is implemented both on distributed memory and shared memory parallel computers, based on the spitting of latitude zones. This parallel ocean general circulation model is the first one in China.
(4) On the efficient method for the dynamical core of the new generation multi-scale forecasting model
i) We present a new multi-level sparse approximate inverse preconditnioner for the complicated 3-D helmholtz equations in the new generation weather forecasting model. As a result, the new sparse approximate inverse preconditioned GCR and GMRES algorithms are given and successfully applied in the dynamical core. Numerical tests show that the new algorithms perform very efficiently, and can greatly improve the efficiency of numerical model.
本文从气象预报数值模式分布式存储并行计算的实际需要出发，着眼于谱元素方法和有限差分的并行计算性能，研究了二维半隐谱元素浅水波模式、中国科学院大气物理研究所第三代海洋环流模式和中国新一代多尺度预报模式等三个气象预报模式的高效并行计算，设计实现了相应的应用软件。本文的主要研究成果概括如下：
(1) 讨论了数值并行计算技术的发展现状及其对数值预报模式的影响，深入研究了并行计算模型、并行编程模型和数值并行软件的软件工程等问题。
(2) 关于谱元素方法及两维半隐谱元素浅水波模式的并行计算
ⅰ) 依据切比契夫多项式和勒让德多项式的计算特点，提出了一个混合局部基函数预备定理，给出了两类混合局部基谱元素格式。采用混合局部基函数可以在不降低计算精度的情况下使元素矩阵具有特定的稀疏特征，且能保持谱元素方法在元素界面上的C~0连续性。
ⅱ) 将有限元方法中的逐元素(EBE)策略应用于谱元素方法的并行计算，在PC机群上设计实现了并行应用软件PSES。研究表明：即使在网络性能瓶颈非常突出的以太网PC机群上，采用高阶谱元素方法(高于7阶)和通常的空间离散规模(32×32个元素以上)求解两维椭圆问题，可实现60％以上的8机并行效率。
ⅲ) 研究了两维半隐谱元素浅水波模式的高效并行计算，给出了离散方程的逐元素(EBE)并行求解技巧，设计了相应的并行应用软件，并在PC机群上获得高效实现。
(3) 关于中科院大气物理研究所第三代海洋环流模式的高效并行计算
ⅰ) 基于对有限差分离散的分析，提出了一个面向显式时间积分并行计算的多重叠
国防科学技术大学研究生院学位论文
边界优化模型，指出了网络速度与处理器速度间的落差使得采用冗余计算取代
部分通信成为一种具有相对优势的实用技术。结合以矩阵矢量乘积为主的迭代
计算特点，从通信量、通信计算比和并行可扩展性等方面比较分析了离散格式
对于并行计算的影响。
11)发展了国内第一个高效并行海洋环流模式:深入研究了中国科学院大气物理所
第三代海洋环流模式，分析了其串行计算特点和并行可扩展性，指出了其计算
方法上存在的对并行计算的某些限制，对该模式组织了有效并行计算，并成功
应用了包括多重叠边界在内的若干并行优化方法，在国产小型MPP系统和SGI
originZoo。上都获得了较高的性能。
(4)关于中国新一代多尺度预报模式动力内核的高效计算问题
i)三维复杂核姆霍兹方程的离散求解是中国新一代多尺度预报模式动力内核中
的一个瓶颈问题，该问题的计算性能决定了整个数值模式的效率。对此，本文
提出了一种基于逐层门限技术的近似逆矩阵稀疏模式预选方法，并构造了相应
的稀疏近似逆预条件子，结合GCR算法和G州[R卫S算法，首次将逐层门限稀
疏近似逆预条件子应用于新一代多尺度预报模式动力内核的实际计算，数值实
验表明这里给出的方法可以大大提高数值模式的计算效率。
The development of atmosphere science, especially the numerical meteorologic forecasting model, and the parallel computing are closed linked. To meet the practical requirements for distributed-memory parallel computing of numerical forecasting models, we study on the parallel characteristics of spectral element and finite difference methods. On this basis, we develop fast and parallel computations for three meteorologic models, including 2-D shallow water equations, the new generation multi-scale weather forecasting model and the ocean general circulation model. The main work is summarized as follows:
(1) We analyze the situation of the parallel computing technology and it's influence on numerical forecasting models, and point out that the numerical models now are in serious demand for efficient parallel algorithms. We also review parallel computing model, parallel programming model and the software engineering for numerical applications.
(2) On spectral element method and parallel computing of 2-D spectral element shallow water model
i) We put forward a hybrid-local-basis theorem, and give two kinds of local basis schemes fitting for both the Chebyshev polynomials and the Legendre case. With these spectral element schemes, we can get elemental matrix with specific sparse pattern, without sacrificing precision. In addition, the famous C?continuity can be hold by these schemes.
ii) The parallel computing of spectral element methods by element-by-element strategy is analyzed deeply, and we design a high-performance spectral element parallel software package named PSES. Numerical tests show that the high-order (>7) spectral element methods can performance efficiently on the relatively slow-network PC clusters, and parallel efficiency with eight nodes is more than 60% with normal elemental scale (32*32 elements).
iii) For the semi-implicit spectral element method on the 2-D shallow water equations, we give a parallel mass-matrix-diagonal preconditioned CG method with the element-by-element strategy and develop the corresponding parallel software package. We also get high efficiency on PC clusters.
(3) On parallel computing of the third generation Ocean General Circulation Model from LASG/IAP
i) We present an optimization model by multi-overlapping-boundary for parallel explicit integration with finite difference discretization, and point out that the gap between speed of network and CPU makes it a practical technique. We compare the parallel characteristics of spectral element and finite difference from the point of view of communications, communication-to-computation ratio and scalability.
ii) We organize the parallel computing of'the third generation ocean general circulation model from the Chinese Institute of Atmosphere Physics (LASG/IAP), and we also apply several parallel optimization strategies including multi-overlapping-boundary. The parallelism is implemented both on distributed memory and shared memory parallel computers, based on the spitting of latitude zones. This parallel ocean general circulation model is the first one in China.
(4) On the efficient method for the dynamical core of the new generation multi-scale forecasting model
i) We present a new multi-level sparse approximate inverse preconditnioner for the complicated 3-D helmholtz equations in the new generation weather forecasting model. As a result, the new sparse approximate inverse preconditioned GCR and GMRES algorithms are given and successfully applied in the dynamical core. Numerical tests show that the new algorithms perform very efficiently, and can greatly improve the efficiency of numerical model.
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